(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a system with a constant number of particles.

Write the total differential dS in terms of the derivarives [itex]\frac{∂S}{∂T}[/itex] and [itex]\frac{∂S}{∂V}[/itex]. Introduce C_{V}(calorific capacity at constant volume).

Next write the total differential of the volume dV in terms of the parcial derivatives [itex]\frac{∂V}{∂T}[/itex] and [itex]\frac{∂V}{∂P}[/itex]. Assume that the pressure is constant. Show that the result comes in the form of:

C_{P}-C_{V}= Expression

2. Relevant equations

C_{P}=T[itex]\frac{∂S}{∂T}[/itex] , P and N Constant

C_{V}=T[itex]\frac{∂S}{∂T}[/itex] , V and N Constant

3. The attempt at a solution

First I wrote the differential of the entropy as asked:

dS = [itex]\frac{∂S}{∂T}[/itex]dT + [itex]\frac{∂S}{∂V}[/itex]dV

I know that [itex]\frac{∂S}{∂V}[/itex] = C_{V}/T. Substituting I get:

dS = C_{V}/T dT + [itex]\frac{∂S}{∂V}[/itex]dV

Next I found the differential of the volume:

dV = [itex]\frac{∂V}{∂T}[/itex]dT + [itex]\frac{∂V}{∂P}[/itex]dP

Since the pressure is constant it reduces to the form

dV = [itex]\frac{∂V}{∂T}[/itex]dT

Substituting in our dS expression we get:

dS = C_{V}/T dT + [itex]\frac{∂S}{∂V}[/itex][itex]\frac{∂V}{∂T}[/itex]dT =

= C_{V}/T dT + [itex]\frac{∂S}{∂T}[/itex]dT =

= C_{V}/T dT + C_{P}/T dT ⇔

⇔ T[itex]\frac{dS}{dT}[/itex] = C_{V}+ C_{P}

I'm making some mistake. If anyone could point me in the right direction I'd appreciate.

Thanks!

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# Homework Help: Thermodynamics - Calorific Capacity

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